Download Modeling Asset Prices in Continuous Time

Derivatives & Risk Management
•  Lectures 12-13:
• Part VII: Option pricing & Continuous-time Finance
– Modeling Asset Prices in Continuous Time
• Part VIII: Option pricing – Advanced Topics
– Derivation of the Black-Scholes Formula
– Option Pricing w/ varying risk-free rate
Part IX:
Managing Market Risk
•  Also in this Lecture (14):
• Part IX: Option Risk Management
– The “Greeks”
Managing Market Risk
•  Stochastic Process & Price Modeling
• continuous time
• continuous variables
•  Markov Processes
• market efficiency
•  Wiener Processes
•  Ito Processes & Ito’s Lemma
Example
•  A FI has SOLD
for $300,000 a European call option
on 100,000 shares of a non-dividend paying stock:
S = 49
X = 50
r = 5%
σ = 20%
T –t = 20 weeks
µ = 13%
•  The Black-Scholes value
•  How does the FI hedge
of the option is $240,000
its risk?
FI = financial institution
Example 2: Naked & Covered Positions
Naked position
Take NO action
Covered position
Buy 100,000 shares today
Both strategies leave the FI exposed to
significant risk, e.g. ST = 40, ST = 60.
Example 3: Stop-Loss Strategy?
This involves:
•  Fully covering the option as soon as it moves
in the money
•  Staying naked the rest of the time
This “deceptively simple” hedging strategy
does NOT work well! Underlying must be
bought slightly above X and sold slightly
below X.
Delta Hedging
•  Idea:
-  Use the Black-Scholes insight
-  to construct a risk-free portfolio
Delta Hedging 2: Definition
•  Delta (Δ) is the rate of change of
the option price with respect to
the underlying
•  Figure 13.2 (p. 286) & 17.2 (p. 361)
Slope = Δ
B
A
Delta Hedging 3: Formulas
- N(-d1) = [N (d 1) – 1]
•  The delta of a European call on a stock paying
dividends at rate q is
N (d 1)e – q(T-t)
•  The delta of a European put is
e – q(T-t) [N (d 1) – 1]
•  See Figure 17.3 or 18.3 (call) and 17.4 or 18.4 (put)
• Delta of call always between zero and 1.
• Delta of put always between -1 and 0.
•  See also Excel spreadsheets
• DerivaGem
•  Delta changes most dramatically when S = X
•  As T-t gets small:
• ITM deltas go up
• OTM deltas go down (why?)
Delta of p
Delta of c
0.0000
-0.2000 1
-0.4000
1.2000
1.0000
0.8000
0.6000
0.4000
0.2000
0.0000
Stock price
Delta Hedging 4: Plots
N (d 1)
•  The delta of a European put is
8
15 22 29 36
43 50 57
64 71 78 85
-0.6000
-0.8000
-1.0000
1
8
15 22 29 36 43 50 57 64 71 78 85 92 99
Stock Price
∂ƒ
∂S
Option
price
• buying delta shares
• for every option which is written.
•  The delta of a European call on a stock is
Δ=
-1.2000
Stock Price
92 99
Delta Hedging 5: Hedging
Delta of Call Options
•  This involves maintaining a delta neutral portfolio
1.2
1
0.8
0.6
0.4
0.2
0
[see TABLES 17.2-3 (H7&7) & 18.2-3 (H8) ]
ITM
ATM
0.99
0.92
0.85
0.78
0.71
0.64
0.5
0.57
0.43
0.36
0.29
0.22
0.15
0.08
0.01
OTM
Time to Maturity
Example: Table 17.2-3 or 18.2-3, Wk 9
•  B-S option value initially is $240,000. In week 9
the value is $414,500, so the FI has lost $174,500
on the option.
•  The FI has also borrowed $1,442,700 in cash.
•  But the shares have gone from $2,557,800 to
$4,171,100 during the nine weeks.
•  The net effect is a loss of $3,900
•  When you rebalance infinitely often, the loss will
be zero.
•  The hedge position must be frequently rebalanced
•  Delta hedging a written option involves a
“BUY high, SELL low” trading rule
This is the cost of the hedge
(in addition to transactions costs)
Using Futures to Hedge Options
•  Futures could have different maturity, T*
F = S exp(r(T*-t))
•  For hedging
ΔS = ΔF exp(-r(T*-t))
•  The required futures position is therefore
HF = exp(-r(T*-t)) HA
where HA = req’d position in the underlying (S)
•  Use (r-q) for r when dividends are paid
Theta
Delta of a Portfolio
•  The delta of a portfolio of options is just the
weighted sum of the individual deltas (why?)
n
Δ = ∑ w i Δi
i =1
•  The weights wi equal the number of underlying
option contracts written.
•  Theta (θ) of a derivative
(or portfolio of derivatives)
is the rate of change of its value with respect to the
passage of time
∂ƒ
θ=
∂t
•  See Figure 15.5
(H7; 18.5 in H8) for the variation of θ
with respect to the stock price for a European call
•  We do not hedge against the passage of time!
• BUT, θ is useful to describe relationships between other
hedge parameters.
Gamma
Gamma
•  Gamma (Γ) is the rate of change of delta (Δ) with
respect to the price of the underlying
∂ 2ƒ
Γ=
∂S 2
0.0800
0.0600
0.0400
•  Gamma indicates how often we should rebalance.
•  See Figure 15.9 (H7; 18.9 in H8) for the variation of Γ
0.0200
0.0000
1
8
15 22 29 36
with respect to the stock price for a call or put
option
43 50 57
64 71 78 85
92 99
Stock Price
Gamma 3: Interpretation
Gamma 2: Why do we need it?
•  Idea: Gamma Addresses Delta Hedging Errors
Caused By
Curvature
•  Figure 17.7Call(p. 369)
•  For a delta neutral portfolio (e.g. short 1 call and
long Δ shares), a Taylor series expansion series
shows that ΔΠ ≈ θΔt + ½ΓΔS 2
ΔΠ
ΔΠ
price
ΔS
C´´
ΔS
C´
C
S
S´
Stock price
Gamma 4:
Making a Portfolio Gamma-Neutral
•  As stocks and futures have zero Gamma, you need
to buy/sell traded options to change Gamma
•  Gamma of delta neutral portfolio plus traded
option is Γ + wT ΓT, so set wT = - Γ/ ΓT
•  Changing traded option position changes delta, so
adjust for this by changing stock position.
Positive Gamma
Negative Gamma
Gamma 5: Example
•  Suppose a portfolio is delta neutral and has a
gamma of -3,000. The delta and gamma of a
particular traded option is 0.62 and 1.50
respectively.
•  Make portfolio gamma neutral by buying
3,000/1.5 = 2,000 options.
•  This changes delta from 0 to 0.62*2,000 = 1,240.
So sell 1,240 shares of underlying to regain delta
neutrality.
Relationship Among
Delta, Gamma, & Theta
Gamma 6: Formulae
•  For a European call or put w/ dividends
Γ=
Recall the B-S differential equation
∂ƒ
∂ƒ
∂ 2ƒ
+ rS + ½ σ 2 S 2 2 = r ƒ
∂t
∂S
∂S
N ' (d1 )e − q (T −t )
Sσ T − t
• Option on FX: q = rf
• Option on Future: S = F, and q = r.
•  N’(d) is bell-curve:
For a non - dividend paying stock
θ + rS Δ + ½σ 2S2 Γ = r ƒ
2
N ' (d ) =
⎛ d
1
exp⎜⎜ −
2π
⎝ 2
⎞
⎟⎟
⎠
Thus, when delta is 0, and theta is large and positive,
gamma tends to be negative
Vega
Vega
•  Vega (ν) is the rate of change of a derivatives
portfolio with respect to volatility
ν =∂ƒ
15.0000
10.0000
∂σ
•  See Figure 15.11
for the variation of ν with
respect to the stock price for a call or put option
(p. 361)
•  In reality, volatility does seem to change over time so
we might want to hedge that risk away.
•  For a European call or put w/ dividends
Vega = S T − t N ' (d1 )e
0.0000
1
8
15 22 29 36 43 50 57 64 71 78 85 92 99
Stock Price
•  As stocks and futures have zero Vega, you need to
− q (T − t )
•  Vegas from stochastic volatility models are similar
to the our B-S approximate Vega.
5.0000
Vega 3:
Making a Portfolio Vega-Neutral
Vega 2: Formulae
• Option on FX: q = rf
• Option on Future: S = F, and q = r.
(13.16) p. 288
buy/sell traded option to change Vega
•  Vega of delta neutral portfolio plus traded option
is V + wT VT, so set wT = - V / VT
•  Changing the traded option position changes delta,
so adjust for this by changing stock position
Gamma- and Vega-Neutral: Example
•  Consider a delta neutral portfolio with a
• gamma = - 5,000; vega = - 8,000.
•  Suppose a traded option has
• gamma = 0.5; vega = 2.0;
delta = 0.6.
•  Suppose another traded option has
• gamma = 0.8; vega = 1.2;
delta = 0.5
Example [cont.]
•  For joint neutrality, we need
- 5,000 + 0.5*w1 + 0.8*w2 = 0, and
- 8,000 + 2.0*w1 + 1.2*w2 = 0
•  Solving, yields w1 = 400, and w2 = 6,000.
•  Now, delta is 400*0.6 + 6,000*0.5 = 3,240.
•  Thus, sell 3,240 units of the underlying to
reestablish delta neutrality.
Managing Delta, Gamma, & Vega
• 
Δ can be changed by taking a position in the
underlying
• 
To adjust Γ & ν it is necessary to take a
position in an option or other derivative
Rho
Vega
•  Rho is the rate of change of the value of a
derivative with respect to the interest rate
rho =
∂ƒ
∂r
•  For currency options there are 2 rhos
Scenario Analysis
Scenario Analysis [cont.]
•  Scenario analysis & the calculation of value at risk
•  Delta and Gamma are used to calculated VaR’s of
(VaR) is a complement to Δ , Γ , ν, etc.
•  Typical VaR question: What loss level are we
99% certain will not be exceeded over the next 10
days?
•  Download JP Morgan’ RiskMetrics from
http://www.riskmetrics.com
portfolios including options.
•  Stress testing involves checking scenarios with
extreme values for the variables (e.g. FX rate and
volatility).
•  Monte Carlo Simulation involves generating
thousands of scenarios and compute the histogram
of returns across simulations. VaR can then be
found as the e.g. 5th percentile.
Hedging vs
Creation of an Option Synthetically
•  When we are hedging we take positions
that offset Δ , Γ , ν, etc.
•  When we create an option synthetically
we take positions that match Δ, Γ, & ν
Portfolio Insurance [cont.]
•  As the value of the portfolio increases, the Δ of the
put becomes less negative & the position in the
portfolio is increased
•  As the value of the portfolio decreases, the Δ of
the put becomes more negative & more of the
portfolio must be SOLD
•  The strategy did NOT work well on October 19,
1987...
Portfolio Insurance
•  In October of 1987 many portfolio managers
attempted to create put options on their portfolios by
matching Δ
•  This involves initially SELLING enough of the
portfolio (or of index futures) to match the Δ of the
put option
•  Might work well if options markets are not very
liquid and/or desired strikes are not traded.